Advanced Linear Algebra, Lecture 2.4: The four subspaces

Advanced Linear Algebra, Lecture 2.4: The four subspaces

In this lecture, we will give a brief review of some key concepts from undergraduate linear algebra, in preparation for upcoming lectures were we abstract concepts such as the transpose, and learn how to encode a general linear map with a matrix. Though this is not necessary, it will help give some context for objects and results which may otherwise seem abstract and unfounded. Every matrix has four fundamental subspaces: the column space, the row space, the nullspace, and the left null space. In this lecture, we review these and show how they fit together. We also see four ways to multiply matrices: (i) rows by columns, (ii) by columns, (iii) by rows, and (iv) columns by rows. Finally, we do an example of Gaussian elimination to see how this relates to the four subspaces.

[fsaldan]

I believe in 30'08, columns multiplying rows, i and j should be equal, they cannot vary independently. So it's not a double sum.

[thomashoffmann]

Around 13:00: should on the left side be written: "col span RAT" or "row span RA" ?
Otherwise the "row span RAT" would be the same as "col span RA" (?) 🤔

[dacianbonta]

Professor Macauley falling prey to the siren song of numericalism. What's 4 subspaces' point at this stage of conceptual development?

Politechnik quantitivism finally got to'im. Well, the intention was good.

[dacianbonta]

I mean there is the tensorial POV for this matter.

I mean,
Render thou unto Strang what is of Strang and render thou unto Bourbaki what is of Bourbaki